Whatever the merits of undertaking a line by line critique of the Australian Curriculum, it would take a long time, it would be boring and it would probably overshadow the large, systemic problems. (Also, no one in power would take any notice, though that has never really slowed us down.) Still, the details should not be ignored, and we’ll consider here one of the gems of Homer Simpson cluelessness.
In 2010, Burkard Polster and I wrote an Age newspaper column about a draft of the Australian Curriculum. We focused on one line of the draft, an “elaboration” of Pythagoras’s Theorem:
recognising that right-angled triangle calculations may generate results that can be integral, fractional or irrational numbers known as surds
Though much can be said about this line, the most important thing to say is that it is wrong. Seven years later, the line is still in the Australian Curriculum, essentially unaltered, and it is still wrong.
OK, perhaps the line isn’t wrong. Depending upon one’s reading, it could instead be meaningless. Or trivial. But that’s it: wrong and meaningless and trivial are the only options.
The weird grammar and punctuation is standard for the Australian Curriculum. It takes a special lack of effort, however, to produce phrases such as “right-angled triangle calculations” and “generate results”. Any student who offered up such vague nonsense in an essay would know to expect big red strokes and a lousy grade. Still, we can take a guess at the intended meaning.
Pythagoras’s Theorem can naturally be introduced with 3-4-5 triangles and the like, with integer sidelengths. How does one then obtain irrational numbers? Well, “triangle calculations” on the triangle below can definitely “generate” irrational “results”:
Yeah, yeah, is not a “surd”. But of course we can replace each by √7 or 1/7 or whatever, and get sidelengths of any type we want. These are hardly “triangle calculations”, however, and it makes the elaboration utterly trivial: fractions “generate” fractions, and irrationals “generate” irrationals. Well, um, wow.
We assume that the point of the elaboration is that if two sides of a right-angled triangle are integral then the third side “generated” need not be. So, the Curriculum writers presumably had in mind 1-1-√2 triangles and the like, where integers unavoidably lead us into the world of irrationals. Fair enough. But how, then, can we similarly obtain the promised (non-integral) fractional sidelengths? The answer is that we cannot.
It is of course notable that two sides of a right-angled triangle can be integral with the third side irrational. It is also notable, however, that two integral sides cannot result in the third side being a non-integral fraction. This is not difficult to prove, and makes a nice little exercise; the reader is invited to give a proof in the comments. The reader may also wish to forward their proof to ACARA, the producers of the Australian Curriculum.
How does such nonsense make it into a national curriculum? How does it then remain there, effectively unaltered, for seven years? True, our 2010 column wasn’t on the front of the New York Times. But still, in seven years did no one at ACARA ever get word of our criticism? Did no one else ever question the elaboration to anyone at ACARA?
But perhaps ACARA did become aware of our or others’ criticism, reread the elaboration, and decided “Yep, it’s just what we want”. It’s a depressing thought, but this seems as likely an explanation as any.
9 Replies to “Obtuse Triangles”
Not having read said ACARA documents (I thought the task would be slightly more pointless than a sphere) – my first thoughts upon reading “elaboration of P-T” would mean situations where the squares of two sides summed to less than the square of the third side (or greater than) hence producing either acute or obtuse triangles (which I thought may have been the case given the heading).
Now to work on that proof…
Nice article. Not sure I agree that “surd” should be deleted from the Curriculum.
BTW, “recognising that right-angled triangle calculations may generate results that can be integral, fractional or irrational numbers known as surds” appears in Year 8 on the draft curriculum on p. 120.
Maybe they mean that when you find the third side of a right triangle using the lengths of two sides and PT, the answer might be an integer, a rational number, or an irrational number … which is obviously true.
Give me an example that “results” in a non-integer fraction.
2, 3/2, 5/2
This example is no different from my 3π-4π-5π absurdity. I do not believe this is what ACARA means.
To be fair, ACARA does not make the assumption that the lengths of two sides must be integers.
Perhaps students are so used to getting questions where the answer is an integer or an irrational number, that ACARA is making the point that the answer can be a non-integer fraction.
Jesus, Terry. Occam’s razor. ACARA fucked up. And, after ten years they still either don’t realise or care that they fucked up.